Overlapping self-affine sets of Kakeya type

TL;DR

This paper calculates the Minkowski dimension of a specific class of overlapping self-affine sets inspired by Kakeya sets, providing explicit examples without restrictive conditions.

Contribution

It extends the computation of Minkowski dimension to non-generic, overlapping self-affine sets of Kakeya type without norm restrictions on linear maps.

Findings

Minkowski dimension computed for all sets in the class

Explicit open subsets with overlaps analyzed

No restrictions on linear map norms required

Abstract

We compute the Minkowski dimension for a family of self-affine sets on the plane. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.